3. Theoretical Framework for Analysing Agricultural Land Use
Most analytical studies on agricultural land use and deforestation are based on either subsistence models or open-economy models Kaimowitz and Angelsen, 1998. These
two types of models differ a lot in the underlying assumptions and result in different explanations for agricultural land expansion Angelsen, 1995. In subsistence models
households are assumed to minimize labour to reach a predetermined level of consumption. Markets are assumed to be missing. Such models result in the
‘population approach’: population growth, a poor technology, bad soil characteristics etc. are considered as the major driving forces of deforestation. In open-economy
models households are assumed to maximize profits facing a set of perfect markets. According to this approach -the ‘market approach’- the relative profitability of
agriculture determines the area cultivated and deforested. Hence the two approaches lead to conflicting policy implications and are therefore extremely relevant. However,
the assumptions on household behaviour and the functioning of markets, especially the labour market, in these models are rather unrealistic for the case of rural areas in
developing countries. Therefore we use a Chayanovian model in which a representative household maximizes utility allowing for substitution between
consumption and leisure. This is the standard formulation in the agricultural household literature Singh et al, 1986; Deaton, 1997; Udry, 1997. An analytical
approach using a Chayanovian type model was also developed by Angelsen 1999. However, instead of considering the distance of the agricultural frontier from the
village centre as in Angelsen 1999 we consider the total cultivated area.
Our model further differs from the model in Angelsen 1999 in the specification of the
production function and the assumptions about the functioning of the labour market.
3.1. The Framework
We assume a representative household in the village to maximize utility, a concave function of consumption and leisure;
, l
c U
with ;
lc cl
U U
. The household is involved in agricultural production for which we assume a Cobb Douglas production
function, f , with a productivity shifter. Labour,
L , and land, A , are the factors of production. The marginal return to both factors is positive,
;
A L
f f
and decreasing,
;
AA LL
f f
. The productivity shifter includes land suitability characteristics,
1
S
, and the available technology,
2
S
. The first factor reflects the natural suitability of the area for agricultural cultivation and includes aspects such as slope, altitude, climate, soil
type etc. Better land characteristics and an improved technology increase output. This specification is more realistic than the linear production function with exogenously
given yields used by Angelsen 1999. Following von Thünen 1966 we assume an additional labour cost related to an increased walking time to the fields when more
area is being cultivated and the distance from the village to the fields increases. We specify this cost as a convex function of the agricultural area,
t
xA with
1 t
. The labour market is assumed to be missing. Households depend completely on family
labour and no off-farm employment is possible. We assume that the entire agricultural output is sold in the market at price p and that the resulting income is completely
consumed. The household will decide how much land to cultivate and how to allocate
the total available time, T to leisure and agricultural production. This leads to the following optimisation problem:
Maximize ,
l c
U With
, A
L pf
c =
and
β α
A L
S S
A L
f
2 1
, =
with 1
; ≤
≤ β
α 1
= +
β α
Subject to a time constraint:
t
xA l
L T
+ +
= with
1 t
The Lagrangian function of the maximization problem is ,
t
xA l
L T
l c
U −
− −
+ =
θ ψ
leading to the following first order conditions: =
− =
∂ Ψ
∂ θ
l
U l
1 =
− =
∂ Ψ
∂ θ
L c
pf U
L 2
1
= −
= ∂
Ψ ∂
− t
A c
txA pf
U A
θ 3
From these three equations it follows that:
L c
l
pf U
U =
= θ
4
1 −
=
t A
L
txA pf
pf
5 The Lagrange multiplier
θ can be interpreted as the shadow wage Mas Colell et al, 1995. Equation 4 expresses that the household will allocate its time in such a way
that both marginal utility from leisure and marginal utility from labour in agriculture are equal to the shadow wage. Equation 5 expresses that land and labour will be
used till the marginal return to those factors are equalized. Considering that
β α
A L
S S
A L
f
2 1
, =
it follows from equation 5 that:
t
txA L
β α
= 6
Substituting θ with
l
U and L with expression 6 in equation 3 and rearranging results in:
Y S
pS U
U A
X l
c
=
2 1
with
1 1
1 1
=
− −
− t
t
tx Y
α α
α β
7 and
1 1
1 1
1 −
− =
− −
= t
t X
α β
α Equations 6 and 7 along with the time constraint determine the amount of land the
representative household will cultivate and how the available time will be allocated to labour and leisure.
Assuming that all households in the village have the same preferences we can aggregate equation 7 over all households in the village
5
and obtain an expression for the total village agricultural area,
agr A
_ :
Y S
pS U
U H
HA agr
A
X l
c
= =
2 1
_ 8
with H the number of households in the village; and with X and Y as defined in equation 7.
3.2. The Effect of Exogenous Changes
Kategori